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Theorem limsupreuz 40287
 Description: Given a function on the reals, its supremum limit is real if and only if two condition holds: 1. there is a real number that is smaller or equal than the function, infinitely often; 2. there is a real number that is larger or equal than the function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
limsupreuz.1 𝑗𝐹
limsupreuz.2 (𝜑𝑀 ∈ ℤ)
limsupreuz.3 𝑍 = (ℤ𝑀)
limsupreuz.4 (𝜑𝐹:𝑍⟶ℝ)
Assertion
Ref Expression
limsupreuz (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)))
Distinct variable groups:   𝑘,𝐹,𝑥   𝑗,𝑍,𝑘,𝑥
Allowed substitution hints:   𝜑(𝑥,𝑗,𝑘)   𝐹(𝑗)   𝑀(𝑥,𝑗,𝑘)

Proof of Theorem limsupreuz
Dummy variables 𝑖 𝑙 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2793 . . . 4 𝑙𝐹
2 limsupreuz.2 . . . 4 (𝜑𝑀 ∈ ℤ)
3 limsupreuz.3 . . . 4 𝑍 = (ℤ𝑀)
4 limsupreuz.4 . . . . 5 (𝜑𝐹:𝑍⟶ℝ)
54frexr 39917 . . . 4 (𝜑𝐹:𝑍⟶ℝ*)
61, 2, 3, 5limsupre3uzlem 40285 . . 3 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦)))
7 breq1 4688 . . . . . . . . 9 (𝑦 = 𝑥 → (𝑦 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑙)))
87rexbidv 3081 . . . . . . . 8 (𝑦 = 𝑥 → (∃𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙)))
98ralbidv 3015 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙)))
10 fveq2 6229 . . . . . . . . . . 11 (𝑖 = 𝑘 → (ℤ𝑖) = (ℤ𝑘))
1110rexeqdv 3175 . . . . . . . . . 10 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙)))
12 nfcv 2793 . . . . . . . . . . . . 13 𝑗𝑥
13 nfcv 2793 . . . . . . . . . . . . 13 𝑗
14 limsupreuz.1 . . . . . . . . . . . . . 14 𝑗𝐹
15 nfcv 2793 . . . . . . . . . . . . . 14 𝑗𝑙
1614, 15nffv 6236 . . . . . . . . . . . . 13 𝑗(𝐹𝑙)
1712, 13, 16nfbr 4732 . . . . . . . . . . . 12 𝑗 𝑥 ≤ (𝐹𝑙)
18 nfv 1883 . . . . . . . . . . . 12 𝑙 𝑥 ≤ (𝐹𝑗)
19 fveq2 6229 . . . . . . . . . . . . 13 (𝑙 = 𝑗 → (𝐹𝑙) = (𝐹𝑗))
2019breq2d 4697 . . . . . . . . . . . 12 (𝑙 = 𝑗 → (𝑥 ≤ (𝐹𝑙) ↔ 𝑥 ≤ (𝐹𝑗)))
2117, 18, 20cbvrex 3198 . . . . . . . . . . 11 (∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
2221a1i 11 . . . . . . . . . 10 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2311, 22bitrd 268 . . . . . . . . 9 (𝑖 = 𝑘 → (∃𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∃𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2423cbvralv 3201 . . . . . . . 8 (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
2524a1i 11 . . . . . . 7 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑥 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
269, 25bitrd 268 . . . . . 6 (𝑦 = 𝑥 → (∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗)))
2726cbvrexv 3202 . . . . 5 (∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ↔ ∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗))
28 breq2 4689 . . . . . . . . 9 (𝑦 = 𝑥 → ((𝐹𝑙) ≤ 𝑦 ↔ (𝐹𝑙) ≤ 𝑥))
2928ralbidv 3015 . . . . . . . 8 (𝑦 = 𝑥 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
3029rexbidv 3081 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥))
3110raleqdv 3174 . . . . . . . . . 10 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥))
3216, 13, 12nfbr 4732 . . . . . . . . . . . 12 𝑗(𝐹𝑙) ≤ 𝑥
33 nfv 1883 . . . . . . . . . . . 12 𝑙(𝐹𝑗) ≤ 𝑥
3419breq1d 4695 . . . . . . . . . . . 12 (𝑙 = 𝑗 → ((𝐹𝑙) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
3532, 33, 34cbvral 3197 . . . . . . . . . . 11 (∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3635a1i 11 . . . . . . . . . 10 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑘)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
3731, 36bitrd 268 . . . . . . . . 9 (𝑖 = 𝑘 → (∀𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
3837cbvrexv 3202 . . . . . . . 8 (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
3938a1i 11 . . . . . . 7 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑥 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
4030, 39bitrd 268 . . . . . 6 (𝑦 = 𝑥 → (∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
4140cbvrexv 3202 . . . . 5 (∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)
4227, 41anbi12i 733 . . . 4 ((∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦) ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥))
4342a1i 11 . . 3 (𝜑 → ((∃𝑦 ∈ ℝ ∀𝑖𝑍𝑙 ∈ (ℤ𝑖)𝑦 ≤ (𝐹𝑙) ∧ ∃𝑦 ∈ ℝ ∃𝑖𝑍𝑙 ∈ (ℤ𝑖)(𝐹𝑙) ≤ 𝑦) ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)))
446, 43bitrd 268 . 2 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥)))
45 nfv 1883 . . . . . . . 8 𝑖(𝐹𝑗) ≤ 𝑥
46 nfcv 2793 . . . . . . . . . 10 𝑗𝑖
4714, 46nffv 6236 . . . . . . . . 9 𝑗(𝐹𝑖)
4847, 13, 12nfbr 4732 . . . . . . . 8 𝑗(𝐹𝑖) ≤ 𝑥
49 fveq2 6229 . . . . . . . . 9 (𝑗 = 𝑖 → (𝐹𝑗) = (𝐹𝑖))
5049breq1d 4695 . . . . . . . 8 (𝑗 = 𝑖 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥))
5145, 48, 50cbvral 3197 . . . . . . 7 (∀𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∀𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥)
5251rexbii 3070 . . . . . 6 (∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑘𝑍𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥)
5352rexbii 3070 . . . . 5 (∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥)
5453a1i 11 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥))
55 nfv 1883 . . . . 5 𝑖𝜑
564adantr 480 . . . . . 6 ((𝜑𝑖𝑍) → 𝐹:𝑍⟶ℝ)
57 simpr 476 . . . . . 6 ((𝜑𝑖𝑍) → 𝑖𝑍)
5856, 57ffvelrnd 6400 . . . . 5 ((𝜑𝑖𝑍) → (𝐹𝑖) ∈ ℝ)
5955, 2, 3, 58uzub 39971 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘𝑍𝑖 ∈ (ℤ𝑘)(𝐹𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑖𝑍 (𝐹𝑖) ≤ 𝑥))
60 eqcom 2658 . . . . . . . . . 10 (𝑗 = 𝑖𝑖 = 𝑗)
6160imbi1i 338 . . . . . . . . 9 ((𝑗 = 𝑖 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)))
62 bicom 212 . . . . . . . . . 10 (((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥) ↔ ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
6362imbi2i 325 . . . . . . . . 9 ((𝑖 = 𝑗 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥)))
6461, 63bitri 264 . . . . . . . 8 ((𝑗 = 𝑖 → ((𝐹𝑗) ≤ 𝑥 ↔ (𝐹𝑖) ≤ 𝑥)) ↔ (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥)))
6550, 64mpbi 220 . . . . . . 7 (𝑖 = 𝑗 → ((𝐹𝑖) ≤ 𝑥 ↔ (𝐹𝑗) ≤ 𝑥))
6648, 45, 65cbvral 3197 . . . . . 6 (∀𝑖𝑍 (𝐹𝑖) ≤ 𝑥 ↔ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)
6766rexbii 3070 . . . . 5 (∃𝑥 ∈ ℝ ∀𝑖𝑍 (𝐹𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)
6867a1i 11 . . . 4 (𝜑 → (∃𝑥 ∈ ℝ ∀𝑖𝑍 (𝐹𝑖) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥))
6954, 59, 683bitrd 294 . . 3 (𝜑 → (∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥 ↔ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥))
7069anbi2d 740 . 2 (𝜑 → ((∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∃𝑘𝑍𝑗 ∈ (ℤ𝑘)(𝐹𝑗) ≤ 𝑥) ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)))
7144, 70bitrd 268 1 (𝜑 → ((lim sup‘𝐹) ∈ ℝ ↔ (∃𝑥 ∈ ℝ ∀𝑘𝑍𝑗 ∈ (ℤ𝑘)𝑥 ≤ (𝐹𝑗) ∧ ∃𝑥 ∈ ℝ ∀𝑗𝑍 (𝐹𝑗) ≤ 𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Ⅎwnfc 2780  ∀wral 2941  ∃wrex 2942   class class class wbr 4685  ⟶wf 5922  ‘cfv 5926  ℝcr 9973   ≤ cle 10113  ℤcz 11415  ℤ≥cuz 11725  lim supclsp 14245 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-ico 12219  df-fz 12365  df-fzo 12505  df-fl 12633  df-ceil 12634  df-limsup 14246 This theorem is referenced by:  limsupreuzmpt  40289  limsupgtlem  40327
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